Molino theory for matchbox manifolds

Dyer, Jessica; Hurder, Steven; Lukina, Olga

doi:10.2140/pjm.2017.289.91

Cited by 21 publications

(87 citation statements)

References 58 publications

(221 reference statements)

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“…However, the notion of the Ellis group does not require finite generation, and finite generation was not used in any of the proofs in . One easily checks that Theorem and all results in which we use in this paper are true for countably generated groups as well. However, some related results on strong quasi‐analyticity in may require the finite (more precisely, compact) generation assumption.…”

Section: Introductionmentioning

confidence: 98%

“…One such invariant, called the asymptotic discriminant , has been introduced by the author joint with Hurder in , as a culmination of a series of papers on actions with non‐trivial isotropy groups joint with Dyer and Hurder . The asymptotic discriminant of a minimal equicontinuous action

(X, G, Φ)

is an invariant of return equivalence of minimal Cantor systems.…”

Section: Introductionmentioning

confidence: 99%

“…The relationship between the cardinality of

{\bar{Φ (G)}}_{x}

and the properties of the action was studied in . In particular, the isotropy group

{\bar{Φ (G)}}_{x}

is trivial if and only if the subgroups in the group chain

{false{G_{n} false}}_{n ⩾ 0}

can be chosen to be normal (such actions are called

G

‐odometers in ).…”

Section: Introductionmentioning

confidence: 99%

“…That is, the cardinality of

D_{x}

reflects the global properties of the action. The idea on how to use the Ellis group to develop a local invariant stems from and was fully developed in the joint paper with Hurder . The idea is as follows.…”

Section: Introductionmentioning

confidence: 99%

“…An uncountable family of distinct wild actions of subgroups of

S L (k, Z)

for

k ⩾ 3

was constructed in . In , every finite group and every separable profinite group were realized as discriminant groups of stable actions of subgroups of

S L (k, Z)

k ⩾ 3

. In those examples, given a finite or a separable profinite group

D

, we constructed an action of a subgroup of

S L (k, Z)

such that for every

n ⩾ 0

in the sequence we have

{scriptD}_{x}^{n} ≅ D

, and every map

ψ_{n, n + 1}

is an isomorphism.…”

Section: Introductionmentioning

confidence: 99%

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Arboreal Cantor actions

Lukina

2018

J. London Math. Soc.

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In this paper, we consider minimal equicontinuous actions of discrete countably generated groups on Cantor sets, obtained from the arboreal representations of absolute Galois groups of fields. In particular, we study the asymptotic discriminant of these actions. The asymptotic discriminant is an invariant obtained by restricting the action to a sequence of nested clopen sets, and studying the isotropies of the enveloping group actions in such restricted systems. An enveloping (Ellis) group of such an action is a profinite group. A large class of actions of profinite groups on Cantor sets is given by arboreal representations of absolute Galois groups of fields. We show how to associate to an arboreal representation an action of a discrete group, and give examples of arboreal representations with stable and wild asymptotic discriminant.

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Section: Introductionmentioning

confidence: 98%

(X, G, Φ)

is an invariant of return equivalence of minimal Cantor systems.…”

Section: Introductionmentioning

confidence: 99%

“…The relationship between the cardinality of

{\bar{Φ (G)}}_{x}

and the properties of the action was studied in . In particular, the isotropy group

{\bar{Φ (G)}}_{x}

is trivial if and only if the subgroups in the group chain

{false{G_{n} false}}_{n ⩾ 0}

can be chosen to be normal (such actions are called

G

‐odometers in ).…”

Section: Introductionmentioning

confidence: 99%

“…That is, the cardinality of

D_{x}

Section: Introductionmentioning

confidence: 99%

“…An uncountable family of distinct wild actions of subgroups of

S L (k, Z)

for

k ⩾ 3

was constructed in . In , every finite group and every separable profinite group were realized as discriminant groups of stable actions of subgroups of

S L (k, Z)

k ⩾ 3

. In those examples, given a finite or a separable profinite group

D

, we constructed an action of a subgroup of

S L (k, Z)

such that for every

n ⩾ 0

in the sequence we have

{scriptD}_{x}^{n} ≅ D

, and every map

ψ_{n, n + 1}

is an isomorphism.…”

Section: Introductionmentioning

confidence: 99%

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Arboreal Cantor actions

Lukina

2018

J. London Math. Soc.

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Growth and Structure of Equicontinuous Foliated Spaces

Moreira Galicia

2024

Springer Proceedings in Mathematics &Amp; Statistics

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Growth and homogeneity of matchbox manifolds

Dyer

Hurder

Lukina

2017

Indagationes Mathematicae

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Abstract. A matchbox manifold with one-dimensional leaves which has equicontinuous holonomy dynamics must be a homogeneous space, and so must be homeomorphic to a classical Vietoris solenoid. In this work, we consider the problem, what can be said about a matchbox manifold with equicontinuous holonomy dynamics, and all of whose leaves have at most polynomial growth type? We show that such a space must have a finite covering for which the global holonomy group of its foliation is nilpotent. As a consequence, we show that if the growth type of the leaves is polynomial of degree at most 3, then there exists a finite covering which is homogeneous. If the growth type of the leaves is polynomial of degree at least 4, then there are additional obstructions to homogeneity, which arise from the structure of nilpotent groups.

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Molino theory for matchbox manifolds

Cited by 21 publications

References 58 publications

Arboreal Cantor actions

Arboreal Cantor actions

Growth and Structure of Equicontinuous Foliated Spaces

Growth and homogeneity of matchbox manifolds

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